Model of Immune Response with Time-Dependent Immune Reactivity

The probability of getting a disease is related to the efficiency of the immune system, which can change with the seasons of the year. This Demonstration shows the solution of a model of the immune system that has periodic changes in the immune reactivity due to changes in the environment.

The model consists of three delay ordinary differential equations

,

,

,

with initial history functions and ; is the immune reactivity (the immune response of the infected individual): , and is the antibody production rate per plasma cell due to the presence of antigens. Here , , and represent antigen, plasma cells, and antibody concentrations; is the antigen reproduction rate; is the probability of an antigen-antibody encounter; is the reciprocal of the plasma cell lifetime; is the number of antibodies necessary to suppress one antigen; and is a constant. Values of these parameters are taken from the reference. Time is , is the time delay necessary for the formation of plasma cells and antibodies, and is the length of the season. Large ratios of antibody to antigen concentrations, , correspond to a strong immune system in which reactions are fast and in which the organism has strong resistance; on the other hand, small values of this ratio imply immunodeficiency.